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There is a problem due tio the fact so many machines with different demands can be plugged in to a 50A circuit, some allow a 12 wire and some an 8. A true 50 for a range or oven a 6 but it uses a different recept.

what size breaker

You are probable plugging this into a 30 amp or a 50 amp circuit and leave it plugged in most of the time. I cannot duplicate the math for the 14 guage wire, so I might stay on the safe side and use a 10 gauge cord. My concern is that a 14 gauge wire could burn pretty good in a fault and not trip a 30 amp breaker or a 50 amp if you use a typical welding circuit. It would take a direct short to do it. You can shop for a 10 ga 120 volt 3 wire extension cord and change both ends. Sometimes cheaper. I might say that every time I built a 50' cord I found it too short.

You're good

You're fine. In the house circuit you need at least 8 ga. Wire and a 40 amp breaker. At a minimum you need 14 ga for 15 amp circuits, 12 ga for 20 amp circuits, 10 ga for 30 amp circuits, and 8 ga. for 40 amps. You should use a 40 amp breaker for your 30 amp (power input max) welder.

Not really

The purpose of a breaker is to protect the wire, not the extension cord or the welder. You MUST size your wire for the max power the breaker will allow to pass. That means if you are using 12 gage wire nothing greater than 20 amps draw. Your welder draws nearly 30 amps, so 12 gauge wire is INADEQUATE and presents a definite fire hazard and wouldn't be allowed by any inspector in the land. If you are going to pull 30 amps (more or less) from the circuit the minimum breaker you will require will be a 40 amperage, and it will require 8/3 wire in this case. Also, because there is no neutral connection in the receptacle you will be using and because the ground wire is normally smaller than the other wires in the cable you will want to substitute the neutral wire for the (bare) ground wire you will need to identify having done so by putting a band of green tape on each end of your white wire. Otherwise you run the very likely risk of frying the wire and the possibility of burning the place to the ground.

Do not skimp on wire size! It is at best false economy and always a safety hazard.

The OP has not posted again here... there is lots of "you could get away with..." stuff posted but why bother? If you spend the money on a nice welder why risk burning the shop to the ground with a small extension cord? If you can get a 50' 8/3 cord for a good price, and you can with the internet those days, then get it. Then you can run any 50 amp circuit device on that cord. Even if you only have a small welder now who know what you will want or need in the future? Spend the money once for quality and then you will Benin the clear!